Method of performing cell search for a wireless communications system

ABSTRACT

Performing cell search in a wireless communications system includes receiving a preamble signal, match filtering the preamble signal with a pseudo noise sequence to form a filtered preamble signal of a plurality of filtered preamble signals, choosing a largest filtered preamble signal from the plurality of filtered preamble signals, and determining an estimated pseudo noise sequence index and an estimated integer part frequency offset according to the largest filtered preamble signal.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to cell search for mobile wirelesssystems, and more particularly, to a method of performing cell searchand integer part frequency offset estimation for a wirelesscommunications system.

2. Description of the Prior Art

Orthogonal frequency division multiple access (OFDMA) has become one ofthe most promising technologies in modern wireless communicationssystems, owing to its robustness against frequency-selective channels,its flexibility for multi-rate transmission considering multiple users,and so on. It is also adopted as one of the air interfaces in IEEE802.16, known as WirelessMAN-OFDMA.

WirelessMAN-OFDMA is a connection-oriented network, in which each framehas a preamble, a downlink subframe, and an uplink subframe, for timedivision duplex (TDD) mode operation. The preamble is an OFDM symbolwith a cyclic prefix (CP) extension like other OFDM symbols within theframe. The difference between the preamble and normal OFDM symbols isthat the preamble is binary-phase shift keying (BPSK) modulated by 114possible pseudo-noise (PN) sequences transmitted by base stations (BSs).Mobile stations (MSs) detect the transmitted PN sequence among the 114possibilities, so that the basic information of the BS, such as itssegment index and cell number, may be acquired to perform demodulationof the downlink subframes. The procedure used to detect the PN sequenceis called “cell search.”

Although the selected set of PN sequences is characterized by low crosscorrelation, presence of cross correlation values can not be ignoredwhen detecting the employed PN sequence, and therefore the computationpower required to perform cell search is high. In addition, whenconsidering the presence of integer part frequency offset, theuncertainty increases, such that the computation burden becomes evenheavier.

SUMMARY OF THE INVENTION

According to a first embodiment of the present invention, a method ofperforming cell search in a wireless communications system includesreceiving a preamble signal, match filtering the preamble signal with afirst pseudo noise sequence to form a first filtered preamble signal,match filtering the preamble signal with a second pseudo noise sequenceto form a second filtered preamble signal, modifying the second filteredpreamble signal to form a modified filtered preamble signal, summing atleast the first filtered preamble signal with the modified filteredpreamble signal to form one of a plurality of summed preamble signals,choosing a largest summed preamble signal from the plurality of summedpreamble signals, determining an estimated pseudo noise sequence indexand an estimated integer part frequency offset according to the largestsummed preamble signal, matching filtering the preamble signal with atleast a first pseudo noise sequence and a second pseudo noise sequencecorresponding to the estimated pseudo noise sequence index and theestimated integer part frequency offset to form a plurality of filteredpreamble signals, and generating an estimated pseudo noise sequence froma largest filtered preamble signal of the plurality of filtered preamblesignals. Optimum solution for integer part frequency offset and pseudonoise sequence index based on maximum a posteriori probability is:

${\left( {{\hat{f}}_{i},\hat{P}} \right) = {\arg{\max\limits_{f_{i},P^{(i)}}{\sum\limits_{l = 0}^{L - 1}\frac{{{\sum\limits_{n = 0}^{N - 1}{{r^{*}\lbrack n\rbrack}{{\hat{s}}_{l,f_{i}}\lbrack n\rbrack}}}}^{2}}{{\hat{C}}_{l}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}}}}},{{where}\text{:}}$${{s_{l,f_{i}}\lbrack n\rbrack} \equiv {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{\left( {{P^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{- \frac{j\; 2\pi\;{kl}}{N}}} \right){\mathbb{e}}^{j\frac{2\pi{({k + f_{i}})}n}{N}}}}}},\left\{ {{{\begin{matrix}{{{\hat{s}}_{0,f_{i}}\lbrack n\rbrack} \equiv {s_{0,f_{i}}\lbrack n\rbrack}} & {l = 0} \\{{{\hat{s}}_{l,f_{i}}\lbrack n\rbrack} \equiv {{s_{l,f_{i}}\lbrack n\rbrack} - {\sum\limits_{\alpha = 0}^{l - 1}{{{\hat{s}}_{\alpha,f_{i}}\lbrack n\rbrack}\frac{\eta\left( {l,\alpha} \right)}{{\hat{C}}_{\alpha}^{2} + \frac{\sigma^{2}}{\sigma_{\alpha}^{2}}}}}}} & {{l \neq 0},}\end{matrix}{\eta\left( {\alpha,\beta} \right)}} \equiv {\sum\limits_{n = 0}^{N - 1}{{s_{\alpha,f_{i}}\lbrack n\rbrack}{{\hat{s}}_{\beta,f_{i}}^{*}\lbrack n\rbrack}}}},{{C_{0}^{2} \equiv {E\left\{ {{P\lbrack k\rbrack}}^{2} \right\}}} = \sigma_{s}^{2}},{{and}\left\{ \begin{matrix}{{\hat{C}}_{0}^{2} \equiv C_{0}^{2}} & {l = 0} \\{{\hat{C}}_{l}^{2} \equiv {C_{0}^{2} - {\sum\limits_{\alpha = 0}^{l - 1}\frac{{{\eta\left( {l,\alpha} \right)}}^{2}}{{\hat{C}}_{\alpha}^{2} + \frac{\sigma^{2}}{\sigma_{\alpha}^{2}}}}}} & {l \neq 0.}\end{matrix} \right.}} \right.$

According to a second embodiment of the present invention, a method ofperforming cell search in a wireless communications system comprisesreceiving a preamble signal, match filtering the preamble signal with apseudo noise sequence to form a filtered preamble signal of a pluralityof filtered preamble signals, choosing a largest filtered preamblesignal from the plurality of filtered preamble signals, and determiningan estimated pseudo noise sequence index and an estimated integer partfrequency offset according to the largest filtered preamble signal.Optimum solution for integer part frequency offset and pseudo noisesequence index based on maximum a posteriori probability is:

${\left( {{\hat{f}}_{i},\hat{P}} \right) = {\arg{\max\limits_{f_{i},P^{(i)}}{\sum\limits_{l = 0}^{L - 1}\frac{{{\sum\limits_{n = 0}^{N - 1}{{r^{*}\lbrack n\rbrack}{{\hat{s}}_{l,f_{i}}\lbrack n\rbrack}}}}^{2}}{{\hat{C}}_{l}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}}}}},{{where}\text{:}}$${{s_{l,f_{i}}\lbrack n\rbrack} \equiv {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{\left( {{P^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{{- j}\frac{\;{2\pi\;{kl}}}{N}}} \right){\mathbb{e}}^{j\frac{2\pi{({k + f_{i}})}n}{N}}}}}},\left\{ {{{\begin{matrix}{{{\hat{s}}_{0,f_{i}}\lbrack n\rbrack} \equiv {s_{0,f_{i}}\lbrack n\rbrack}} & {l = 0} \\{{{\hat{s}}_{l,f_{i}}\lbrack n\rbrack} \equiv {{s_{l,f_{i}}\lbrack n\rbrack} - {\sum\limits_{\alpha = 0}^{l - 1}{{{\hat{s}}_{\alpha,f_{i}}\lbrack n\rbrack}\frac{\eta\left( {l,\alpha} \right)}{{\hat{C}}_{\alpha}^{2} + \frac{\sigma^{2}}{\sigma_{\alpha}^{2}}}}}}} & {{l \neq 0},}\end{matrix}{\eta\left( {\alpha,\beta} \right)}} \equiv {\sum\limits_{n = 0}^{N - 1}{{s_{\alpha,f_{i}}\lbrack n\rbrack}{{\hat{s}}_{\beta,f_{i}}^{*}\lbrack n\rbrack}}}},{{C_{0}^{2} \equiv {E\left\{ {{P\lbrack k\rbrack}}^{2} \right\}}} = \sigma_{s}^{2}},{{and}\left\{ \begin{matrix}{{\hat{C}}_{0}^{2} \equiv C_{0}^{2}} & {l = 0} \\{{\hat{C}}_{l}^{2} \equiv {C_{0}^{2} - {\sum\limits_{\alpha = 0}^{l - 1}\frac{{{\eta\left( {l,\alpha} \right)}}^{2}}{{\hat{C}}_{\alpha}^{2} + \frac{\sigma^{2}}{\sigma_{\alpha}^{2}}}}}} & {l \neq 0.}\end{matrix} \right.}} \right.$

According to a third embodiment of the present invention, a method ofperforming cell search in a wireless communications system comprisesreceiving a preamble signal, match filtering the preamble signal with aplurality pseudo noise sequences to form a plurality of filteredpreamble signals, summing the plurality of filtered preamble signals toform one of a plurality of summed preamble signals, choosing a largestsummed preamble signal from the plurality of summed preamble signals,determining an estimated pseudo noise sequence index and an estimatedinteger part frequency offset according to the largest summed preamblesignal, matching filtering the preamble signal with at least a firstpseudo noise sequence and a second pseudo noise sequence correspondingto the estimated pseudo noise sequence index and the estimated integerpart frequency offset to form a plurality of filtered preamble signals,and generating an estimated pseudo noise sequence from a largestfiltered preamble signal of the plurality of filtered preamble signals.Optimum solution for integer part frequency offset and pseudo noisesequence index based on maximum a posteriori probability is:

${\left( {{\hat{f}}_{i},\hat{P}} \right) = {\arg{\max\limits_{f_{i},P^{(i)}}{\sum\limits_{l = 0}^{L - 1}\frac{{{\sum\limits_{n = 0}^{N - 1}{{r^{*}\lbrack n\rbrack}{{\hat{s}}_{l,f_{i}}\lbrack n\rbrack}}}}^{2}}{{\hat{C}}_{l}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}}}}},{{where}\text{:}}$${{s_{l,f_{i}}\lbrack n\rbrack} \equiv {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{\left( {{P^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{{- j}\frac{\;{2\pi\;{kl}}}{N}}} \right){\mathbb{e}}^{j\frac{2\pi{({k + f_{i}})}n}{N}}}}}},\left\{ {{{\begin{matrix}{{{\hat{s}}_{0,f_{i}}\lbrack n\rbrack} \equiv {s_{0,f_{i}}\lbrack n\rbrack}} & {l = 0} \\{{{\hat{s}}_{l,f_{i}}\lbrack n\rbrack} \equiv {{s_{l,f_{i}}\lbrack n\rbrack} - {\sum\limits_{\alpha = 0}^{l - 1}{{{\hat{s}}_{\alpha,f_{i}}\lbrack n\rbrack}\frac{\eta\left( {l,\alpha} \right)}{{\hat{C}}_{\alpha}^{2} + \frac{\sigma^{2}}{\sigma_{\alpha}^{2}}}}}}} & {{l \neq 0},}\end{matrix}{\eta\left( {\alpha,\beta} \right)}} \equiv {\sum\limits_{n = 0}^{N - 1}{{s_{\alpha,f_{i}}\lbrack n\rbrack}{{\hat{s}}_{\beta,f_{i}}^{*}\lbrack n\rbrack}}}},{{C_{0}^{2} \equiv {E\left\{ {{P\lbrack k\rbrack}}^{2} \right\}}} = \sigma_{s}^{2}},{{and}\left\{ \begin{matrix}{{\hat{C}}_{0}^{2} \equiv C_{0}^{2}} & {l = 0} \\{{\hat{C}}_{l}^{2} \equiv {C_{0}^{2} - {\sum\limits_{\alpha = 0}^{l - 1}\frac{{{\eta\left( {l,\alpha} \right)}}^{2}}{{\hat{C}}_{\alpha}^{2} + \frac{\sigma^{2}}{\sigma_{\alpha}^{2}}}}}} & {l \neq 0.}\end{matrix} \right.}} \right.$

According to a fourth embodiment of the present invention, a method forgenerating an integer part frequency offset set comprises generating afirst sum of magnitudes of sub-carriers whose index is a multiple of 3of the preamble signal, generating a second sum of magnitudes ofsub-carriers whose index is a multiple of 3n+1 of the preamble signal,generating a third sum of magnitudes of sub-carriers whose index is amultiple of 3n+2 of the preamble signal, determining a greatest sum ofthe first, second and third sums, and determining the integer partfrequency offset set corresponding to the greatest sum. A range of n isapproximately one-third number of sub-carriers of the preamble signal.

These and other objectives of the present invention will no doubt becomeobvious to those of ordinary skill in the art after reading thefollowing detailed description of the preferred embodiment that isillustrated in the various figures and drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a table of integer part frequency offsets for different ratiosand signal bandwidth.

FIG. 2 is a frequency-domain diagram of a received OFDM signal of apreamble.

FIG. 3 is a block diagram of an embodiment of a cell search algorithmaccording to the present invention.

FIGS. 4-6 are flowcharts of embodiments of performing cell search in awireless communications system by utilizing the cell search algorithm ofFIG. 3.

FIG. 7 is a flowchart of an embodiment for generating a set of integerpart frequency offsets by utilizing the cell search algorithm of FIG. 3.

FIG. 8 is a diagram of a window for calculating the metric in thepreamble where frame boundary is only coarsely estimated.

FIG. 9 is a diagram of another window for calculating the metric in thepreamble.

FIG. 10 is a diagram of a window for estimating the metric of a noiseterm.

FIG. 11 is a diagram of a threshold for deciding number of combinedmulti-path metrics.

DETAILED DESCRIPTION

A cell search algorithm with ±20 ppm frequency offset of MSs inWirelessMAN-OFDMA of IEEE 802.16 may be considered in one embodiment. Acell search block is activated by a fractional-part frequency offsetestimation, and therefore it is assumed that the fractional-partfrequency offset is negligible. However, integer part frequency offsetuncertainty is large (±9 sub-carriers at 3.8 GHz band over 10 MHz signalbandwidth considering ±2 ppm frequency offset at BS (c.f. Clause8.4.14.1 [1])). The cell search algorithm may cover a range of frequencyoffsets from ±0 ppm to hundreds of ppm.

According to data sheets provided by RF vendors, a ratio of thefrequency offsets of MSs may be less than ±20 ppm. The correspondinginteger part frequency offsets can be found with respect to differentratios of frequency offsets and signal bandwidth (shown in FIG. 1),where the maximum possible integer part frequency offset is denoted byfint sub-carriers.

It is also assumed that coarse frame/symbol synchronization is providedby preamble detection and/or a delayed differential correlator, suchthat the received timing is coarsely known. Because the preamble symbolsare currently not accumulated over frames, only the frequency-selectiveslowly fading channel is considered in the sequel.

Letting s(t) be the baseband transmitted signal of a preamble,

$\begin{matrix}{{{s(t)} = {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{P\lbrack k\rbrack}{\mathbb{e}}^{j\;\frac{2\pi\;{Wkt}}{N}}}}}},} & (1)\end{matrix}$where P[k]ε{0,±1} is the symbol value at sub-carrier k for k=0, 1, . . ., N−1; N is the FFT size of the considered system; and W is thenull-to-null bandwidth of the transmitted signal. Letting f_(i) ε Z bethe integer part frequency offset, the received low-pass equivalentsignal rLP(t) over the frequency-selective slowly fading channel is

$\begin{matrix}{{r_{LP}(t)} = {{\sum\limits_{l = 0}^{L - 1}{\rho_{l}{s\left( {t - \frac{l}{W}} \right)}{\mathbb{e}}^{{j2\pi}\; f_{i}\frac{W}{N}t}}} + {z_{LP}(t)}}} & (2)\end{matrix}$where ρ₀, ρ₁, . . . , ρ_(L−1) are independent zero-mean complex-valuedGaussian random variables with variance σ₀ ², σ₁ ², . . . , σ_(L−1) ²,respectively. By sampling r_(LP)(t) at a rate W.

$\begin{matrix}{{{{r\lbrack n\rbrack} \equiv {r_{LP}\left( \frac{n}{W} \right)}} = {{\frac{1}{N}{\sum\limits_{l = 0}^{L - 1}{\rho_{l}{\sum\limits_{k = 0}^{N - 1}{{P\lbrack k\rbrack}{\mathbb{e}}^{j\;\frac{2\pi\;{k{({n - l})}}}{N}}{\mathbb{e}}^{j\;\frac{2\pi\; f_{i}n}{N}}}}}}} + {z\lbrack n\rbrack}}}{{{{for}\mspace{14mu} n} = 0},1,\ldots\mspace{14mu},{N - 1.}}} & (3)\end{matrix}$

Assuming that the integer part frequency offset and the employed PNsequence are independent and uniformly distributed, a maximum aposteriori probability (MAP) detection rule is

${\left( {{\hat{f}}_{i},\hat{P}} \right) = {\arg\;{\max\limits_{f_{i},P^{(i)}}{\Pr\left( {{{r\lbrack n\rbrack};{n = 0}},1,\ldots\mspace{14mu},\left. {N - 1} \middle| P^{(i)} \right.,f_{i}} \right)}}}},$where P^((i))=(P^((i))[0],P^((i))[1], . . . ,P^((i))[N−1]) is an ith PNsequence and

Pr (r[n]; n = 0, n = 1, …  , N − 1|P^((i)), f_(i)) = ∫∫  …  ∫Pr (r[n]; n = 0, 1, …  , N − 1|P^((i)), f_(i), ρ₀, ρ₁, …  , ρ_(L − 1)) × Pr (ρ₀, ρ₁, …  , ρ_(L − 1))𝕕ρ₀𝕕ρ₁  …  𝕕ρ_(L − 1).

Then,

$\left( {{\hat{f}}_{i},\hat{P}} \right) = {\arg\;{\max\limits_{f_{i},P^{(i)}}{\sum\limits_{l = 0}^{L - 1}\frac{{{\sum\limits_{n = 0}^{N - 1}{{r^{*}\lbrack n\rbrack}{{\hat{s}}_{l,f_{i}}\lbrack n\rbrack}}}}^{2}}{{\hat{C}}_{l}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}}}}$where${{s_{l,f_{i}}\lbrack n\rbrack} \equiv {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{\left( {{P^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{{- j}\;\frac{2\pi\;{kl}}{N}}} \right){\mathbb{e}}^{j\;\frac{2\pi{({k + f_{i}})}n}{N}}}}}},\left\{ {{{\begin{matrix}{{{\hat{s}}_{0,f_{i}}\lbrack n\rbrack} \equiv {s_{0,f_{i}}\lbrack n\rbrack}} & {l = 0} \\{{{\hat{s}}_{l,f_{i}}\lbrack n\rbrack} \equiv {{s_{l,f_{i}}\lbrack n\rbrack} - {\sum\limits_{\alpha = 0}^{l - 1}{{{\hat{s}}_{\alpha,f_{i}}\lbrack n\rbrack}\frac{\eta\left( {l,\alpha} \right)}{{\hat{C}}_{\alpha}^{2} + \frac{\sigma^{2}}{\sigma_{\alpha}^{2}}}}}}} & {{l \neq 0},}\end{matrix}{\eta\left( {\alpha,\beta} \right)}} \equiv {\sum\limits_{n = 0}^{N - 1}{{s_{\alpha,f_{i}}\lbrack n\rbrack}{{\hat{s}}_{\beta,f_{i}}^{*}\lbrack n\rbrack}}}},{{C_{0}^{2} \equiv {E\left\{ {{P\lbrack k\rbrack}}^{2} \right\}}} = \sigma_{s}^{2}},\left\{ \begin{matrix}{{\hat{C}}_{0}^{2} \equiv C_{0}^{2}} & {l = 0} \\{{\hat{C}}_{l}^{2} \equiv {C_{0}^{2} - {\sum\limits_{\alpha = 0}^{l - 1}\frac{{{\eta\left( {l,\alpha} \right)}}^{2}}{{\hat{C}}_{\alpha}^{2} + \frac{\sigma^{2}}{\sigma_{\alpha}^{2}}}}}} & {l \neq 0.}\end{matrix} \right.} \right.$

The superscript * stands for the conjugation operation. By ignoring thesmall value on the auto-correlation function η(α,β) when α≠β and

${{{\alpha - \beta}}{\operatorname{<<}\frac{N}{3}}},$the suboptimal but simplified form becomes

$\begin{matrix}{\left( {{\hat{f}}_{i},\hat{P}} \right) = {\arg\;{\max\limits_{f_{i},P^{(i)}}{\sum\limits_{l = 0}^{L - 1}{\frac{{{\sum\limits_{n = 0}^{N - 1}{{r^{*}\lbrack n\rbrack}{s_{l,f_{i}}\lbrack n\rbrack}}}}^{2}}{\sigma_{s}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}.}}}}} & (4)\end{matrix}$

According to Parseval's formula,

${{{\sum\limits_{n = 0}^{N - 1}{{r^{*}\lbrack n\rbrack}{s_{l,f_{i}}\lbrack n\rbrack}}}}^{2} = {{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R^{*}\lbrack k\rbrack}{S_{l,f_{i}}\lbrack k\rbrack}}}}}^{2}},$where

${R\lbrack k\rbrack} \equiv {\sum\limits_{n = 0}^{N - 1}{{r\lbrack n\rbrack}{\mathbb{e}}^{{- j}\;\frac{2\pi\;{kn}}{N}}\mspace{14mu}{and}\mspace{14mu}{S_{l,f_{i}}\lbrack k\rbrack}}} \equiv {\sum\limits_{n = 0}^{N - 1}{{s_{l,f_{i}}\lbrack n\rbrack}{\mathbb{e}}^{{- j}\;\frac{2\pi\;{kn}}{N}}}}$are the N-point FFTs of r[n] and s[n], respectively. Letting

${{s\lbrack n\rbrack} \equiv {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{P^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{j\;\frac{2\pi\;{kn}}{N}}}}}},{{s_{l,f_{i}}\lbrack n\rbrack} = {{s\left\lbrack \left( {n - l} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\;\frac{2\pi\; f_{i}n}{N}}}}$and${S_{l,f_{i}}\lbrack k\rbrack} = {{P^{(i)}\left\lbrack \left( {k - f_{i}} \right)_{N} \right\rbrack}{{\mathbb{e}}^{{- j}\;\frac{2\pi{({k - f_{i}})}l}{N}}.}}$

Therefore, the suboptimal realization in (4) becomes

$\begin{matrix}{{\left( {{\hat{f}}_{i},\hat{P}} \right) = {\arg\;{\max\limits_{f_{i},P^{(i)}}{\mu\left( {f_{i},P^{(i)}} \right)}}}}{where}\begin{matrix}{{\mu\left( {f_{i},P^{(i)}} \right)} = {\sum\limits_{l = 0}^{L - 1}\frac{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R^{*}\lbrack k\rbrack}{S_{l,f_{i}}\lbrack k\rbrack}}}}}^{2}}{\sigma_{s}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}}} \\{= {\sum\limits_{l = 0}^{L - 1}\frac{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R^{*}\lbrack k\rbrack}{P^{(i)}\left\lbrack \left( {k - f_{i}} \right)_{N} \right\rbrack}{\mathbb{e}}^{{- j}\;\frac{2\pi{({k - f_{i}})}l}{N}}}}}}^{2}}{\sigma_{s}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}}} \\{= {\sum\limits_{l = 0}^{L - 1}\frac{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(i)}\left\lbrack \left( {k - f_{i}} \right)_{N} \right\rbrack}{\mathbb{e}}^{{j2\pi}\;{{kl}/N}}}}}}^{2}}{\sigma_{s}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}}}\end{matrix}} & (5)\end{matrix}$

Defining T^((i))[k]≡R[k]P^((i))[(k−f_(i))_(N)],

$\begin{matrix}{{{\mu\left( {f_{i},P^{(i)}} \right)} = {\sum\limits_{l = 0}^{L - 1}{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{T^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{j\; 2\pi\;{{kl}/N}}}}}}^{2} \cdot \xi_{l}}}},} & (6)\end{matrix}$where ξ₁=σ_(l) ²/(σ_(l) ²σ_(s) ²+σ²).

As pointed out by Kuang-jen Wang and Mao-Ching Chiu, the term

$\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{T^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{j\; 2\pi\;{{kl}/N}}}}$in (6) is actually the N-point IFFT of T^((i))[k]. Therefore, letting

${{t^{(i)}\lbrack l\rbrack} \equiv {{1/N} \cdot {\sum\limits_{k = 0}^{N - 1}{{T^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{j\; 2\pi\;{{kl}/N}}}}}},$

$\begin{matrix}{{\mu\left( {f_{i},P^{(i)}} \right)} = {\sum\limits_{l = 0}^{L - 1}{{{t^{(i)}\lbrack l\rbrack}}^{2} \cdot {\xi_{l}.}}}} & (7)\end{matrix}$

In addition, please note that P^((i))[(k−f_(i))_(N)]ε{0, ±1} for kε{0,1,. . . , N−1}, and hence calculation of T^((i))[k] does not involvemultiplication.

Computing the metric for each possible integer part frequency offset andeach possible PN sequence, the number of IFFT operations required is114×(2[Δf_(max)/df]+3), which is very large, e.g. 114×21=2166. It is,however, possible to reduce number of candidates of integer partfrequency offsets and PN sequences.

To reduce the candidates of integer part frequency offsets, the receivedtime-domain signal is considered:

$\begin{matrix}{{r\lbrack n\rbrack} = {{\frac{1}{N}{\sum\limits_{l = 0}^{L - 1}{\rho_{l}{\sum\limits_{k = 0}^{N - 1}{{P\lbrack k\rbrack}{\mathbb{e}}^{{j2\pi}\;{{k{({n - l})}}/N}}{\mathbb{e}}^{2\pi\; f_{i}{n/N}}}}}}} + {z\lbrack n\rbrack}}} \\{= {{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{\left( {\sum\limits_{l = 0}^{L - 1}{\rho_{l}{P\lbrack k\rbrack}{\mathbb{e}}^{{- {j2\pi}}\;{{kl}/N}}}} \right){\mathbb{e}}^{2{\pi{({k + f_{i}})}}{n/N}}}}} + {{z\lbrack n\rbrack}.}}}\end{matrix}$

The received frequency-domain signal is then

$\begin{matrix}{{R\lbrack k\rbrack} = {\sum\limits_{n = 0}^{N - 1}{{r\lbrack n\rbrack}{\mathbb{e}}^{{- j}\; 2\;\pi\;{{nk}/N}}}}} \\{= {{\frac{1}{N}{\sum\limits_{l = 0}^{L - 1}{\rho_{l}{\sum\limits_{n = 0}^{N - 1}{\sum\limits_{k^{\prime} = 0}^{N - 1}{{P\left\lbrack k^{\prime} \right\rbrack}{\mathbb{e}}^{j\; 2\;{\pi{({k - k^{\prime} + f_{i}})}}{n/N}}{\mathbb{e}}^{{- 2}\;\pi\;{{kl}/N}}}}}}}} + {Z\lbrack k\rbrack}}} \\{= {{\frac{1}{N}{\sum\limits_{l = 0}^{L - 1}{\rho_{l}{\sum\limits_{k^{\prime} = 0}^{N - 1}{{P\left\lbrack k^{\prime} \right\rbrack}\left( {\sum\limits_{n = 0}^{N - 1}{\mathbb{e}}^{j\; 2\;{\pi{({k - k^{\prime} + f_{i}})}}{n/N}}} \right){\mathbb{e}}^{{- 2}\;\pi\;{{kl}/N}}}}}}} + {Z\lbrack k\rbrack}}} \\{= {{\sum\limits_{l = 0}^{L - 1}{\rho_{l}{P\left\lbrack \left( {k - f_{i}} \right)_{N} \right\rbrack}{\mathbb{e}}^{{- 2}\;\pi\;{{kl}/N}}}} + {Z\lbrack k\rbrack}}} \\{= {{{H\lbrack k\rbrack}{P\left\lbrack \left( {k - f_{i}} \right)_{N} \right\rbrack}} + {Z\lbrack k\rbrack}}}\end{matrix}$ where${Z(k)} = {\sum\limits_{n = 0}^{N - 1}{{z\lbrack n\rbrack}{\mathbb{e}}^{{- j}\; 2\;\pi\;{{nk}/N}}}}$and${H(k)} = {{\sum\limits_{l = 0}^{L - 1}{\rho_{l}{\mathbb{e}}^{{- j}\; 2\;\pi\;{{nl}/N}}}} = {\sum\limits_{n = 0}^{N - 1}{\rho_{n}{\mathbb{e}}^{{- j}\; 2\;\pi\;{{nk}/N}}}}}$since it is assumed that ρ_(n)=0 for n≧L.

For any PN sequence P, P[k]=0 for k≠−426+s(mod3)≡s(mod3) where s is asegment index of P. So,

${R\lbrack k\rbrack} = {{{{H\lbrack k\rbrack}{P\left\lbrack \left( {k - f_{i}} \right)_{N} \right\rbrack}} + {Z\lbrack k\rbrack}}\mspace{45mu} = \left\{ \begin{matrix}{{{{H\lbrack k\rbrack}{P\left\lbrack \left( {k - f_{i}} \right)_{N} \right\rbrack}} + {Z\lbrack k\rbrack}},} & {{k \equiv {f_{i} + {s\left( {{mod}\; 3} \right)}}};} \\{{Z\lbrack k\rbrack},} & {{otherwise}.}\end{matrix} \right.}$

FIG. 2 shows the received frequency-domain signal. In FIG. 2, it can beseen that the received signal is composed of noise in a carrier k, wherek≠f_(i)+s(mod3). Hence, a scheme may be proposed to reduce the number ofcandidates of the integer part frequency offsets. First,

$\begin{matrix}{\sum\limits_{k \equiv {{seg}{({{mod}\; 3})}}}{{R\lbrack k\rbrack}}^{2}} & (8)\end{matrix}$is computed for seg=0,1,2, and

${seg}^{*} = {\arg\;{\max_{seg}{\sum\limits_{k \equiv {{seg}{({{mod}\; 3})}}}{{R\lbrack k\rbrack}}^{2}}}}$is denoted. Then, only the integer part frequency offsets f_(i) forwhichf _(i) +s=seg*(mod3)are considered. Let f_(d)=f_(i)+s. Since f_(i)+sε{−f_(int), −f_(int)+1,. . . ,f_(int)+1, f_(int)+2}, the number of candidates of f_(d) handledis reduced to [(2f_(int), +3)/3]. Let P^((i)) be an ith PN sequence withsegment index s^((i)). The metric in (5) may then be written as

$\begin{matrix}{{\mu\left( {f_{i},P^{(i)}} \right)} = {{\sum\limits_{l = 0}^{L - 1}\frac{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R^{*}\lbrack k\rbrack}{P^{(i)}\left\lbrack \left( {k - f_{d} + s^{(i)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}}}}}^{2}}{\sigma_{s}^{2} + \frac{\sigma^{2}}{\sigma_{l}^{2}}}}\mspace{104mu} = {{\sum\limits_{l = 0}^{L - 1}{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{T^{(i)}\lbrack k\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}}}}}^{2} \cdot \xi_{l}}}\mspace{104mu} = {\sum\limits_{l = 0}^{L - 1}{{{t^{(i)}\lbrack l\rbrack}}^{2} \cdot \xi_{l}}}}}} & (9)\end{matrix}$where T^((i))[k]=R*[k]P^((i))[(k−f_(d)+s^((i)))_(N)] and ξ₁=σ₁ ²/(σ_(l)²σ_(s) ²+σ²). Now, this metric needs only be computed whenf_(d)≡seg*(mod3). Thus, for a given f_(d), the metric for one of the 114PN sequences may computed by performing one IFFT operation.

Next, a method is provided for computing the metric for more than one PNsequence in only one IFFT operation. First, Np PN sequences are addeddirectly. LetQ ^((i) ^(g) ⁾ =P ^((i) ^(g) ^(N) ^(P) ⁾ +P ^((i) ^(g) ^(N) ^(P) ⁺¹⁾ + .. . +P ^((i) ^(g) ^(N) ^(P) ^(+N) ^(P) ⁻¹⁾   (10)andΦ^((i) ^(g) ⁾ [k]=R[k]Q ^((i) ^(g) ⁾[(k−f _(i))_(N)]  (11)for i_(g)=0,1,2, . . . ,└114/N_(P)┘−1. Corresponding time domain signalscan then be obtained as

${\Phi^{(i_{g})}\lbrack l\rbrack} = {{1/N} \cdot {\sum\limits_{k = 0}^{N - 1}{{\Phi^{(i_{g})}\lbrack k\rbrack}{{\mathbb{e}}^{j\; 2\pi\;{{kl}/N}}.}}}}$

Next, the metric is computed as in (9), and {circumflex over (f)}_(d)and î_(g) corresponding to the largest metric are chosen:

$\left( {{\hat{f}}_{d},{\hat{i}}_{g}} \right) = {{\arg\;{\max\limits_{f_{d},i_{g}}{\mu\left( {f_{d},i_{g}} \right)}}} = {\sum\limits_{l = 0}^{L - 1}{{{\phi^{(i_{g})}\lbrack l\rbrack}}^{2} \cdot {\xi_{l}.}}}}$

Finally, the metric of the Np PN sequences are added in the combinedsequence Q^((i) ^(g) ⁾, respectively. Letting

${\hat{i}}^{\prime} = {\arg\;{\max\limits_{i}{\overset{\sim}{\mu}\left( P^{(i)} \right)}}}$where${{\overset{\sim}{\mu}\left( P^{(i)} \right)} = {\sum\limits_{l = 0}^{L - 1}{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(i)}\left\lbrack \left( {k - {\hat{f}}_{d} + s^{(i)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}}}}}^{2}\xi_{l}}}},$the index of the estimated PN sequence is î_(g) N _(P)+î′. Followingthis process, the total number of IFFT operations to complete the cellsearch algorithm is [114/N_(P)]·[(2f_(int)+3)/3]+N_(P). When N_(P)becomes large, the total number of IFFT operations needed decreases butthe performance is also degraded. Hence, N_(P) should be decided basedon a tradeoff between complexity and performance.

In addition, adding Np PN sequences directly as in (10) is not a goodmethod for combining the PN sequences. The combination in (11) is thusmodified to

$\begin{matrix}{{\Phi^{(i_{g})}\lbrack k\rbrack} = {\sum\limits_{q = 0}^{N_{p} - 1}{{R\left\lbrack \left( {k - {\Delta_{f} \cdot \left\lbrack {q/2} \right\rbrack}} \right)_{N} \right\rbrack}{P^{({{i_{g}N_{p}} + q})}\left\lbrack \left( {k - f_{d} + s^{({{i_{g}N_{p}} + q})} - {\Delta_{f} \cdot \left\lbrack {q/2} \right\rbrack}} \right)_{N} \right\rbrack}j^{{({1 - {({- 1})}^{q}})}/2}}}} & (12)\end{matrix}$where Δ_(f) is a carrier shift for combining the PN sequences. We alsomultiply the PN sequences with odd index with a factor j.

For example, the combination in (12) is illustrated for Np=4, and thevalue of Δ_(f) is decided. For Np=4, (12) can be expressed as

Φ^((i_(g)))[k] = R[k]P^((4 i_(g)))⌊(k − f_(d) + s^((4 i_(g))))_(N)⌋ + R[k]P^((4 i_(g) + 1))⌊(k − f_(d) + s^((4 i_(g) + 1)))_(N)⌋j + R[(k − Δ_(f))_(N)]P^((4 i_(g) + 2))⌊(k − f_(d) + s^((4 𝕚_(g) + 2)) − Δ_(f))_(N)⌋ + R[(k − Δ_(f))_(N)]P^((4 i_(g) + 3))⌊(k − f_(d) + s^((4 i_(g) + 3)) − Δ_(f))_(N)⌋j for i_(g)=0,1, . . . ,28. For simplicity, consider the metric fori_(g)=0:

$\begin{matrix}{\begin{matrix}{\mspace{79mu}{{\mu\left( {f_{d},0} \right)} = {\sum\limits_{l = 0}^{L - 1}{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{\Phi^{(0)}\lbrack k\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}}}}}^{2}\xi_{l}}}}} \\{= {\sum\limits_{l = 0}^{L - 1}{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}\left( {{{R\lbrack k\rbrack}{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}} +} \right.}}}}} \\{\left. {{R\lbrack k\rbrack}{P^{(1)}\left\lbrack \left( {k - f_{d} + s^{(1)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{{j\; 2\;\pi\;{{kl}/N}} + {j\;{\pi/2}}}} \right) +} \\{\quad{{{R\lbrack k\rbrack}{P^{(2)}\left\lbrack \left( {k - f_{d} + s^{(2)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;{\pi{({k + \Delta_{f}})}}{l/N}}} +}} \\{{{{R\lbrack k\rbrack}{P^{(3)}\left\lbrack \left( {k - f_{d} + s^{(3)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{{j\; 2\;{\pi{({k + \Delta_{f}})}}{l/N}} + {j\;{\pi/2}}}}}^{2}\xi_{l}} \\{= {\sum\limits_{l = 0}^{L - 1}\left( {{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}}}}}^{2} +} \right.}} \\{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(1)}\left\lbrack \left( {k - f_{d} + s^{(1)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{{j2\pi}\;{{kl}/N}}}}}}^{2} +} \\{{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(2)}\left\lbrack \left( {k - f_{d} + s^{(2)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}}}}}^{2} +} \\{{\left. {{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(3)}\left\lbrack \left( {k - f_{d} + s^{(3)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}}}}}^{2} \right)\xi_{l}} +} \\{\sum\limits_{l = 0}^{L - 1}{\frac{2}{N^{2}}\left( {\sum\limits_{p \neq q}{{Re}\left\{ {A^{(p)}\left( A^{(q)} \right)}^{*} \right\}}} \right)\xi_{l}}} \\{= {{\sum\limits_{l = 0}^{L - 1}{\left( {{{t^{(0)}\lbrack l\rbrack}}^{2} + {{t^{(1)}\lbrack l\rbrack}}^{2} + {{t^{(2)}\lbrack l\rbrack}}^{2} + {{t^{(3)}\lbrack l\rbrack}}^{2}} \right)\xi_{l}}} +}} \\{\sum\limits_{l = 0}^{L - 1}{\frac{2}{N^{2}}\left( {\sum\limits_{p \neq q}{{Re}\left\{ {A^{(p)}\left( A^{(q)} \right)}^{*} \right\}}} \right)\xi_{l}}}\end{matrix}\mspace{79mu}{where}{A^{(p)} = {\sum\limits_{k = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(p)}\left\lbrack \left( {k - f_{d} + s^{(p)}} \right)_{N} \right\rbrack}{{\mathbb{e}}^{{{j2}\;{\pi{({k + {\Delta_{f}{\lbrack{p/2}\rbrack}}})}}{l/N}} + {j\;\pi\;{{({1 - {({- 1})}^{p}})}/4}}}.}}}}} & (13)\end{matrix}$

Consider the cross term Re{A⁽⁰⁾(A⁽¹⁾)*} first

${{Re}\left\{ {A^{(0)}\left( A^{(1)} \right)}^{*} \right\}} = {{{Re}\begin{Bmatrix}{\sum\limits_{k = 0}^{N - 1}{\sum\limits_{k^{\prime} = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}{R^{*}\left\lbrack k^{\prime} \right\rbrack}}}} \\{{P^{(1)}\left\lbrack \left( {k^{\prime} - f_{d} + s^{(1)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{{{- j}\; 2\;\pi\; k^{\prime}{l/N}} - {j\;{\pi/2}}}}\end{Bmatrix}} = {{Re}\begin{Bmatrix}{{- j}{\sum\limits_{k = 0}^{N - 1}{\sum\limits_{k^{\prime} = 0}^{N - 1}{{R\lbrack k\rbrack}{R^{*}\left\lbrack k^{\prime} \right\rbrack}{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}}}}} \\{{P^{(1)}\left\lbrack \left( {k^{\prime} - f_{d} + s^{(1)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;{\pi{({k - k^{\prime}})}}{l/N}}}\end{Bmatrix}}}$

When k≠k′, R[k]R*[k′] is small with respect to R[k]R*[k]=|R[k]|².Therefore, the above equation can be approximated to be

$\begin{matrix}{{{{{Re}\left\{ {A^{(0)}\left( A^{(1)} \right)}^{*} \right\}} \approx {{Re}\begin{Bmatrix}{{- j}{\sum\limits_{k = 0}^{N - 1}{{{R\lbrack k\rbrack}}^{2}{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}}}} \\{P^{(1)}\left\lbrack \left( {k - f_{d} + s^{(1)}} \right)_{N} \right\rbrack}\end{Bmatrix}}} = 0},} & (14)\end{matrix}$since |R[k]|²,P⁽⁰⁾[(k−f_(d)+s⁽⁰⁾)_(N)], and P⁽¹⁾[(k−f_(d)+s(¹⁾)_(N)] areall real for all k=0,1, . . . ,N−1. Then, consider the cross term

${{Re}\left\{ {A^{(0)}\left( A^{(2)} \right)}^{*} \right\}} = {{{Re}\begin{Bmatrix}{\sum\limits_{k = 0}^{N - 1}{\sum\limits_{k^{\prime} = 0}^{N - 1}{{R\lbrack k\rbrack}{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}}}} \\{{\mathbb{e}}^{j\; 2\;\pi\;{{kl}/N}}{R^{*}\left\lbrack k^{\prime} \right\rbrack}{P^{(2)}\left\lbrack \left( {k^{\prime} - f_{d} + s^{(2)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{{- j}\; 2\;{\pi{({k^{\prime} + \Delta_{f}})}}{l/N}}}\end{Bmatrix}} = {{Re}{\begin{Bmatrix}{{\mathbb{e}}^{{- j}\; 2\;\pi\;\Delta_{f}{l/N}}{\sum\limits_{k = 0}^{N - 1}{\sum\limits_{k^{\prime} = 0}^{N - 1}{{R\lbrack k\rbrack}{R^{*}\left\lbrack k^{\prime} \right\rbrack}}}}} \\{{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}{P^{(2)}\left\lbrack \left( {k^{\prime} - f_{d} + s^{(2)}} \right)_{N} \right\rbrack}{\mathbb{e}}^{j\; 2\;{\pi{({k - k^{\prime}})}}{l/N}}}\end{Bmatrix}.}}}$

Similar to (14), the approximated equation becomes

$\begin{matrix}{{{{\sum\limits_{l = 0}^{L - 1}{{Re}\left\{ {A^{(0)}\left( A^{(2)} \right)}^{*} \right\}}} \approx {\sum\limits_{l = 0}^{L - 1}{{Re}\begin{Bmatrix}{{\mathbb{e}}^{{- j}\; 2\;\pi\;\Delta_{f}{l/N}}{\sum\limits_{k = 0}^{N - 1}{{{R\lbrack k\rbrack}}^{2}{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}}}} \\{P^{(2)}\left\lbrack \left( {k - f_{d} + s^{(2)}} \right)_{N} \right\rbrack}\end{Bmatrix}}}} = {{Re}\left\{ {\sum\limits_{l = 0}^{L - 1}{{\mathbb{e}}^{{- j}\; 2\;\pi\;\Delta_{f}{l/N}} \cdot B}} \right\}}}\mspace{79mu}{where}\mspace{79mu}{B = {\sum\limits_{k = 0}^{N - 1}{{{R\lbrack k\rbrack}}^{2}{P^{(0)}\left\lbrack \left( {k - f_{d} + s^{(0)}} \right)_{N} \right\rbrack}{P^{(2)}\left\lbrack \left( {k - f_{d} + s^{(2)}} \right)_{N} \right\rbrack}}}}} & (15)\end{matrix}$is not a function of l. Letting Δ_(f)=N/2, (15) can be rewritten as

$\begin{matrix}\begin{matrix}{{\sum\limits_{l = 0}^{L - 1}{{Re}\left\{ {A^{(0)}\left( A^{(1)} \right)}^{*} \right\}}} = {{Re}\left\{ {B \cdot {\sum\limits_{l = 0}^{L - 1}{\mathbb{e}}^{{- {j\pi}}\; l}}} \right\}}} \\{= \left\{ \begin{matrix}{0,\left\lbrack {{Para}\mspace{14mu} 0100} \right\rbrack} & {{{if}\mspace{14mu} L\mspace{14mu}{is}\mspace{14mu}{even}},} \\{B,\left\lbrack {{Para}\mspace{14mu} 0101} \right\rbrack} & {{otherwise}.}\end{matrix} \right.}\end{matrix} & (16)\end{matrix}$

Without loss of generality, it is assumed that the 0^(th) PN sequenceP⁽⁰⁾ is transmitted. Hence, from (14) and (16), (13) can be rewritten as

$\begin{matrix}{{\mu\left( {f_{d},0} \right)} = {{\sum\limits_{l = 0}^{L - 1}{\left( {{{t^{(0)}\lbrack l\rbrack}}^{2} + {{t^{(1)}\lbrack l\rbrack}}^{2} + {{t^{(2)}\lbrack l\rbrack}}^{2} + {{t^{(3)}\lbrack l\rbrack}}^{2}} \right)\xi_{l}}} +}} \\{\sum\limits_{l = 0}^{L - 1}{\frac{2}{N^{2}}\left( {\sum\limits_{p \neq q}{{Re}\left\{ {A^{(p)}\left( A^{(q)} \right)}^{*} \right\}}} \right)\xi_{l}}} \\{\approx {{\sum\limits_{l = 0}^{L - 1}{{{t^{(0)}\lbrack l\rbrack}}^{2}\xi_{l}}} + {\sum\limits_{l = 0}^{L - 1}{\frac{2}{N^{2}}\left( {\sum\limits_{p \neq q}{{Re}\left\{ {A^{(p)}\left( A^{(q)} \right)}^{*} \right\}}} \right)\xi_{l}}}}} \\{\approx {\sum\limits_{l = 0}^{L - 1}{{{t^{(0)}\lbrack l\rbrack}}^{2}{\xi_{l}.}}}}\end{matrix}$

If the cross correlation of any two distinct PN sequences approximateszero, ξ_(l) approximates a constant for all l, and L is even. From theabove equation, it can be seen that the metric of the combined signalcontaining the transmitted PN sequence is close to the metric of thetransmitted PN sequence.

In the above example, the case for N_(P)=4 is considered, and Δ_(f) isdecided to be N/2 under the assumption that ξ_(l) remains constant fordifferent l. For any N_(P), Δ_(f) may be chosen to be

$\Delta_{f} = \frac{N}{\left\lbrack \frac{N_{p}}{2} \right\rbrack}$which is not the best decision if ξ_(l) is not a constant, but which isan appropriate option for any possible ξ_(l)'s.

In practice, statistics of channel impulse response (CIR) are unknown,so the metric should be simplified to be

${\mu\left( {f_{d},P^{(i)}} \right)} = {\sum\limits_{l = 0}^{L - 1}{{{t^{(i)}\lbrack l\rbrack}}^{2}.}}$

Performance decreases if this metric is used to estimate the integerpart frequency offset and the employed PN sequence:

$\left( {{\hat{f}}_{d},\hat{P}} \right) = {\arg{\max\limits_{f_{d},P^{(i)}}{{\mu\left( {f_{d},P^{(i)}} \right)}.}}}$

It is known that ξ_(l)=σ_(l) ²/(σ_(l) ²σ_(s) ²+σ²)=0. So in (9),|t^((i))[l]|² is added in the metric computation only when σ_(l) ²≠0.Hence, the metric may be modified as

${{\mu\left( {f_{d},P^{(i)}} \right)} = {\sum\limits_{l \in L_{N_{path}}}{{t^{(i)}\lbrack l\rbrack}}^{2}}},$where L_(N) _(peak) ={l:0≦l≦L−1,|t^((i))[l]|² is one of the largestN_(path) metric |t^((i))[m]|² for m=0,1, . . . ,L−1}. Furthermore, |·|can be used to replace the operation |·|² to avoid use ofmultiplication. Also, an operation |·|_(a) is used to approximate theoperation |·|, since |a+jb|=√{square root over(a²+b²)}≈max(a,b)+min(a,b)/2=|a+jb|_(a). Hence, the metric finallyemployed may be

$\begin{matrix}{{\mu\left( {f_{d},P^{(i)}} \right)} = {\sum\limits_{l \in L_{P}}{{t^{(i)}\lbrack l\rbrack}}_{a}}} \\{= {{\sum\limits_{l \in L_{P}}{\max\left( {{{Re}\left\{ {t^{(i)}\lbrack l\rbrack} \right\}},{{Im}\left\{ {t^{(i)}\lbrack l\rbrack} \right\}}} \right)}} + {\frac{1}{2}{\min\left( {{{Re}\left\{ {t^{(i)}\lbrack l\rbrack} \right\}},{{Im}\left\{ {t^{(i)}\lbrack l\rbrack} \right\}}} \right)}}}}\end{matrix}$where L_(N) _(peak) ={l:0≦l≦L−1,|t^((i))[l]|_(a) is one of the largestN_(path) metric |t^((i))[m]|_(a) for m=0,1, . . . , L−1}. (8) can alsobe modified to

$\sum\limits_{k \equiv {{seg}{({{mod}\; 3})}}}{{R\lbrack k\rbrack}}_{a}$for seg=0,1,2, with seg* denoted as

${seg}^{*} = {{argmax}_{seg}{\sum\limits_{k \equiv {{seg}{({{mod}\; 3})}}}{{{R\lbrack k\rbrack}}_{a}.}}}$

Please refer to FIGS. 3-7. FIG. 3 is a block diagram of an embodiment ofthe cell search algorithm for N_(P)=3. FIGS. 4-6 are flowcharts ofembodiments of performing cell search in a wireless communicationssystem by utilizing the cell search algorithm of FIG. 3, and FIG. 7 is aflowchart of an embodiment for generating a set of integer partfrequency offsets by utilizing the cell search algorithm of FIG. 3.Referring first to FIG. 4, a method of performing cell search in awireless communications system includes the following steps:

Step 400: Receive a preamble signal.

Step 401: Match filter the preamble signal with a first pseudo noisesequence to form a first filtered preamble signal.

Step 402: Match filter the preamble signal with a second pseudo noisesequence to form a second filtered preamble signal.

Step 403: Modify the second filtered preamble signal to form a modifiedfiltered preamble signal.

Step 404: Sum at least the first filtered preamble signal with themodified filtered preamble signal to form one of a plurality of summedpreamble signals.

Step 405: Choose a largest summed preamble signal from the plurality ofsummed preamble signals.

Step 406: Determine an estimated pseudo noise sequence index and anestimated integer part frequency offset according to the largest summedpreamble signal.

Step 407: Match filter the delayed preamble signal with at least a firstpseudo noise sequence and a second pseudo noise sequence correspondingto the estimated pseudo noise sequence index and the estimated integerpart frequency offset to form a plurality of filtered preamble signals.

Step 408: Generate an estimated pseudo noise sequence from a largestfiltered preamble signal of the plurality of filtered preamble signals.

Referring back to FIG. 3, a preamble signal r[n] is received (Step 400),then match filtered with a first PN sequence P^((3i) ^(g) ⁾[k−f_(d)] anda second PN sequence, e.g. a PN sequence P^((3i) ^(g) ⁺¹⁾[k−f_(d)] or aPN sequence P^((3i) ^(g) ⁺²⁾[k−f_(d)], to form first and second filteredpreamble signals, respectively (Steps 401-402). The second filteredpreamble signal is modified, e.g. by performing phase rotation of 90° orby performing an N/2 carrier shift, to form a modified filtered preamblesignal (Step 403). Then, at least the first filtered preamble signal andthe modified filtered preamble signal are summed to form one of aplurality of summed preamble signals (Step 404). For example, Steps401-404 may be repeated to cover all possible combinations of PNsequences and integer part frequency offsets, as described above,whereby the plurality of summed preamble signals may be formed. Out ofthe plurality of summed preamble signals, a largest summed preamblesignal is selected (Step 405), and an estimated PN sequence index î_(g)and integer part frequency offset {circumflex over (f)}_(d) aredetermined (Step 406) depending on which summed preamble signal of theplurality of summed preamble signals has the largest value. Theestimated PN sequence index î_(g) may then correspond to a group ofN_(P) PN sequences. The FFT output data of the preamble R(k) for k=0,1,. . . ,N−1 is stored before acquiring the integer part frequency offset{circumflex over (f)}_(d). After the integer part frequency offset{circumflex over (f)}_(d) is estimated, the preamble signal is thenmatch filtered with at least a first PN sequence P^((3î) ^(g)⁾[k−{circumflex over (f)}_(d)] and a second PN sequence P^((3î) ^(g)⁺¹⁾[k−{circumflex over (f)}_(d)] corresponding to the estimated PNsequence index î_(g) and the estimated integer part frequency offset{circumflex over (f)}_(d) to form a plurality of filtered preamblesignals (Step 407). Depending on which of the plurality of filteredpreamble signals is largest, an estimated PN sequence is generated (Step408), thus completing the cell search operation.

Referring to FIG. 5, a method of performing cell search in a wirelesscommunications system includes the following steps:

Step 500: Receive a preamble signal.

Step 501: Match filter the preamble signal with a pseudo noise sequenceto form a filtered preamble signal of a plurality of filtered preamblesignals.

Step 502: Choose a largest filtered preamble signal from the pluralityof filtered preamble signals.

Step 503: Determine an estimated pseudo noise sequence index and anestimated integer part frequency offset according to the largestfiltered preamble signal.

Similar to the method of FIG. 4, the preamble signal r[n] is received(Step 500). Then, the preamble signal r[n] is match filtered with apseudo noise sequence, e.g. P^((3i) ^(g) ⁾[k−f_(d)] to form a filteredpreamble signal of a plurality of filtered preamble signals (Step 501).For example, Step 501 may be repeated to cover all possible combinationsof PN sequences and integer part frequency offsets, as described above,whereby the plurality of filtered preamble signals may be formed. Out ofthe plurality of filtered preamble signals, a largest filtered preamblesignal is chosen (Step 502). Then, according to the largest filteredpreamble signal, an estimated PN sequence index î_(g) and an estimatedinteger part frequency offset {circumflex over (f)}_(d) are determined(Step 503).

Referring to FIG. 6, a method of performing cell search in a wirelesscommunications system includes the following steps:

Step 600: Receive a preamble signal.

Step 601: Match filter the preamble signal with a plurality of pseudonoise sequences to form a plurality of filtered preamble signals.

Step 602: Sum the plurality of filtered preamble signals to form one ofa plurality of summed preamble signals.

Step 603: Choose a largest summed preamble signal from the plurality ofsummed preamble signals.

Step 604: Determine an estimated pseudo noise sequence index and anestimated integer part frequency offset according to the largest summedpreamble signal.

Step 605: Match filter the delayed preamble signal with at least a firstpseudo noise sequence and a second pseudo noise sequence correspondingto the estimated pseudo noise sequence index and the estimated integerpart frequency offset to form a plurality of filtered preamble signals.

Step 606: Generate an estimated pseudo noise sequence from a largestfiltered preamble signal of the plurality of filtered preamble signals.

In the method shown in FIG. 6, after receiving the preamble signal (Step600), the preamble signal is match filtered with a plurality of PNsequences, e.g. PN sequences P^((3i) ^(g) ⁾[k−f₁] P^((3i) ^(g)⁺¹⁾[k−f_(d)], and P^((3i) ^(g) ⁺²⁾[k−f_(d)], to form a plurality offiltered preamble signals (Step 601). The plurality of filtered preamblesignals are summed to form one of a plurality of summed preamble signals(Step 602). Then, out of the plurality of summed preamble signals, alargest summed preamble signal is chosen (Step 603), and based on thelargest summed preamble signal, an estimated PN sequence index î_(g) andan estimated integer part frequency offset {circumflex over (f)}_(d) aredetermined (Step 604). At this point, because the estimated PN sequenceindex î_(g) may represent more than one filtered preamble signal, e.g.three filtered preamble signals, the delayed preamble signal is matchfiltered with at least a first PN sequence, e.g. P^((3î) ^(g)⁾[k−{circumflex over (f)}_(d)], and a second PN sequence, e.g. P^((3î)^(g) ⁺¹⁾[k−{circumflex over (f)}_(d)], corresponding to the estimated PNsequence index î_(g) and the estimated integer part frequency offset{circumflex over (f)}_(d) to form a plurality of filtered preamblesignals (Step 605). Then, the estimated PN sequence is generated from alargest filtered preamble signal of the plurality of filtered preamblesignals (Step 606).

FIG. 7 shows a method of reducing frequency-domain uncertainty forreducing number of times needed for searching for the transmitted PNsequence and the integer part frequency offset. To reducefrequency-domain uncertainty, the integer part frequency offset set maybe found by generating three sums of sub-carriers of the receivedpreamble signal. A first sum of magnitudes of sub-carriers whose indexis a multiple of 3, e.g. sub-carriers 0, 3, 6, 9, and so on, isgenerated (Step 700). A second sum of sub-carriers whose index is amultiple of 3 offset by 1, e.g. sub-carriers 1, 4, 7, 10, and so on, isgenerated (Step 701). And, a third sum of sub-carriers whose index is amultiple of 3 offset by 2, e.g. sub-carriers 2, 5, 8, 11, and so on, isgenerated (Step 702). By determining a greatest sum out of the firstsum, the second sum, and the third sum (Step 703), the integer partfrequency offset set may be found corresponding to the greatest sum(Step 704). For example, if the integer part frequency offset is 8sub-carriers, the preamble signal may only have values in sub-carriers8, 11, 14, and so on, so the greatest sum will be the third sum. Thus,the number of integer part frequency offset candidates to be matchfiltered with the preamble signal may be reduced by ⅔. In the above, nhas a range of approximately ⅓ the number of sub-carriers total in thepreamble signal. The method of FIG. 7 may be incorporated into themethods of FIGS. 4, 5, and 6.

Please refer to FIG. 8, which is a diagram of a window for calculatingthe metric in the preamble where frame boundary is only coarselyestimated. FIG. 8 shows the preamble. Since there exists a timing errorafter coarse timing estimation, an OFDM signal at a position on the CPinterval is collected to avoid inter-symbol interference (ISI) caused bythe timing error. Lengths W and L of the window are due to timinguncertainty and delay spread, respectively. As shown in FIG. 8,

${\mu\left( {f_{d},P^{(i)}} \right)} = {\sum\limits_{l \in L_{N_{path}}}{{t^{(i)}\lbrack l\rbrack}_{a}}}$where L_(N) _(peak) ={l:64−W≦l≦64+W+L−1,|t^((i))[l]|_(a) is one of thelargest N_(path) metrics |t^((i))[m]|_(a) for m=64−W,65−W, . . .,63+W+L}. Total window length is 2W+L.

Please refer to FIG. 9, which is a diagram of another window forcalculating the metric in the preamble. In FIG. 8, only channels withmoderate delay spread are considered. Sometimes, however, channels withlarge delay spread, e.g. SUI−5 channels, may be encountered. In thissituation, the window length W should be modified to be large enough tocover range of multi-path delay spread. But, performance for the AWGNchannel degrades due to the large window length W. Therefore, a methodthat is robust to both AWGN channels and fading channels with largedelay spread is provided.

The window to choose the metric is modified to be L_(N) _(path)={l:T_(shift)−W≦l≦T_(shift)+W+1,|t^((i))[l]|_(a) is one of the largestN_(path) metrics |t^((i))[m]|_(a) for m=T_(shift)−W,T_(shift)−W+1, . . ., T_(shift)+W}. And, the total window length is 2W+1. Such modificationis illustrated in FIG. 9.

Please refer to FIG. 10, which is a diagram of a window for estimatingthe metric of a noise term. As shown in FIG. 10, a window of lengthW_(noise) is chosen for estimating the metric of the noise term. Notonly the length, but also position, of the window should be chosencarefully for estimating the metric of the noise term. As shown in FIG.10, the metric of the noise term is averaged from the positionT_(shift)+N_(CP) to T_(shift)+N_(CP)+W_(noise)−1. That is, the estimatedmetric of the noise term {circumflex over (μ)}_(noise) can be given as:

${\hat{\mu}}_{noise} = {\frac{1}{W_{noise}}{\sum\limits_{t = {T_{shift} + N_{CP}}}^{T_{shift} + N_{CP} + W_{noise} - 1}{{t^{(i)}\lbrack l\rbrack}}_{a}}}$

Then, a threshold μ_(thres)=η·{circumflex over (μ)}_(noise) is set. Ifthe largest metric is larger than the threshold, then the metric

${\mu\left( {f_{d},P^{(i)}} \right)} = {\sum\limits_{l \in L_{N_{path}}}{{t^{(i)}\lbrack l\rbrack}}_{a}}$is not changed. Otherwise, the metric

${\mu\left( {f_{d},P^{(i)}} \right)} = {\sum\limits_{l \in L_{2}}{{t^{(i)}\lbrack l\rbrack}}_{a}}$where only the largest two metrics |t^((i))[l]|_(a) are added may beused. This is shown in FIG. 11. For high SNR, {circumflex over(μ)}_(noise) is small, and so is μ_(thres). Hence, the number of metricsadded in μ(f_(d),P^((i))) is usually N_(path). For low SNR, the numberis usually two. For the AWGN channel, the system usually operates at lowSNR.

Therefore, in the above, when computing the metric of the combinedsignals, the following may be computed:

${\mu\left( {f_{d},i_{g}} \right)} = \left\{ \begin{matrix}{{\sum\limits_{l \in L_{N_{path}}}{{\phi^{(i_{g})}\lbrack l\rbrack}}_{a}},} & {{{if}\mspace{14mu}{\max_{l \in L_{N_{path}}}{{\phi^{(i_{g})}\lbrack l\rbrack}}_{a}}} > {\eta_{1} \cdot {\hat{\mu}}_{noise}}} \\{{\sum\limits_{l \in L_{2}}{{\phi^{(i_{g})}\lbrack l\rbrack}}_{a}},} & {{otherwise},}\end{matrix} \right.$where f_(d)≡seg*(mod3) and for i_(g)=0,1, . . . ,[114/N_(P)]−1.

Likewise, when computing the metric of the N_(P) PN sequences (P^((î)^(g) ^(N) ^(P)) ,P^((î) ^(g) ^(N) ^(P) ⁺¹⁾, . . . ,P(^(((î) ^(g) ^(+1)N)^(P) ⁻¹⁾)), the following may be computed:

${\overset{\sim}{\mu}\left( P^{(i)} \right)} = \left\{ {{{\begin{matrix}{{\sum\limits_{l \in L_{N_{path}}}{{t^{(i)}\lbrack l\rbrack}}_{a}},} & {{{if}\mspace{14mu}{\max_{l \in L_{N_{path}}}{{t^{(i)}\lbrack l\rbrack}}_{a}}} > {\eta_{2} \cdot {\hat{\mu}}_{noise}}} \\{{\sum\limits_{l \in L_{2}}{{t^{(i)}\lbrack l\rbrack}}_{a}},} & {otherwise}\end{matrix}{for}\mspace{14mu} i} = {{\hat{i}}_{g}N_{P}}},{{{\hat{i}}_{g}N_{P}} + 1},\ldots\mspace{14mu},{{{\hat{i}}_{g}N_{P}} + N_{P} - 1.}} \right.$

In conclusion, the cell search algorithm considering non-zero integerpart frequency offset is described. A joint integer part frequencyoffset and transmitted PN sequence estimator are derived based on themaximum-likelihood (ML) criterion. This optimal realization requires theGram-Schmidt procedure to mitigate interference introduced by thenon-impulse-like auto-correlation of the PN sequences, whose complexityis high. Therefore, the suboptimum realization is derived to simplifythe complexity. Core operation of the derived cell search may beimplemented by an IFFT operation to reduce complexity. Methods forreducing the uncertainty of the integer part frequency offset and the PNsequences with small performance degradation are described, such thatthe fast cell search is realized at low cost.

Those skilled in the art will readily observe that numerousmodifications and alterations of the device and method may be made whileretaining the teachings of the invention.

1. A method of performing cell search in a wireless communicationssystem, the method comprising: receiving a preamble signal; matchfiltering the preamble signal with a first pseudo noise sequence to forma first filtered preamble signal; match filtering the preamble signalwith a second pseudo noise sequence to form a second filtered preamblesignal; modifying the second filtered preamble signal to form a modifiedfiltered preamble signal; summing at least the first filtered preamblesignal with the modified filtered preamble signal to form one of aplurality of summed preamble signals; choosing a largest summed preamblesignal from the plurality of summed preamble signals; determining anestimated pseudo noise sequence index and an estimated integer partfrequency offset according to the largest summed preamble signal; matchfiltering the preamble signal with at least a first pseudo noisesequence and a second pseudo noise sequence corresponding to theestimated pseudo noise sequence index and the estimated integer partfrequency offset to form a plurality of filtered preamble signals; andgenerating an estimated pseudo noise sequence from a largest filteredpreamble signal of the plurality of filtered preamble signals.
 2. Themethod of claim 1 further comprising: generating a first sum ofmagnitudes of sub-carriers whose index is a multiple of 3 of thepreamble signal; generating a second sum of magnitudes of 3n+1 channelsof the preamble signal; generating a third sum of magnitudes of 3n+2channels of the preamble signal; determining a greatest sum of thefirst, second and third sums; and determining an integer part frequencyoffset set corresponding to the greatest sum; wherein a range of n isapproximately one-third number of channels of the preamble signal; andwherein receiving the preamble signal is receiving a preamble signaloffset by an integer part frequency offset of the integer part frequencyoffset set.
 3. The method of claim 1 wherein modifying the secondfiltered preamble signal to form the modified filtered preamble signalis phase rotating the second filtered preamble signal to form themodified filtered preamble signal.
 4. The method of claim 1 whereinmodifying the second filtered preamble signal to form the modifiedfiltered preamble signal is carrier shifting the second filteredpreamble signal to form the modified filtered preamble signal.
 5. Themethod of claim 1, further comprising: match filtering the preamblesignal with the first pseudo noise sequence at a plurality of delayvalues to form a plurality of filtered preamble signals; wherein formingthe first filtered preamble signal comprises combining the plurality offiltered preamble signals to form the first filtered preamble signal. 6.A method of performing cell search in a wireless communications system,the method comprising: receiving a preamble signal; match filtering thepreamble signal with a pseudo noise sequence to form a filtered preamblesignal of a plurality of filtered preamble signals, wherein matchfiltering the preamble signal with the pseudo noise sequence to form thefiltered preamble signal of the plurality of filtered preamble signalscomprises match filtering the preamble signal with the pseudo noisesequence at a plurality of delay values to form the plurality offiltered preamble signals; choosing a largest filtered preamble signalfrom the plurality of filtered preamble signals; and determining anestimated pseudo noise sequence index and an estimated integer partfrequency offset according to the largest filtered preamble signal. 7.The method of claim 6 further comprising: generating a first sum ofmagnitudes of 3n channels of the preamble signal; generating a secondsum of magnitudes of 3n+1 channels of the preamble signal; generating athird sum of magnitudes of 3n+2 channels of the preamble signal;determining a greatest sum of the first, second and third sums; anddetermining an integer part frequency offset set corresponding to thegreatest sum; wherein a range of n is approximately one-third number ofchannels of the preamble signal; and wherein receiving the preamblesignal is receiving a preamble signal offset by an integer partfrequency offset of the integer part frequency offset set.
 8. A methodof performing cell search in a wireless communications system, themethod comprising: receiving a preamble signal; match filtering thepreamble signal with a plurality of pseudo noise sequences to form aplurality of filtered preamble signals; summing the plurality offiltered preamble signals to form one of a plurality of summed preamblesignals; choosing a largest summed preamble signal from the plurality ofsummed preamble signals; and determining an estimated pseudo noisesequence index and an estimated integer part frequency offset accordingto the largest summed preamble signal; match filtering the preamblesignal with at least a first pseudo noise sequence and a second pseudonoise sequence corresponding to the estimated pseudo noise sequenceindex and the estimated integer part frequency offset to form aplurality of filtered preamble signals; and generating an estimatedpseudo noise sequence from a largest filtered preamble signal of theplurality of filtered preamble signals.
 9. The method of claim 8 furthercomprising: generating a first sum of magnitudes of 3n channels of thepreamble signal; generating a second sum of magnitudes of 3n+1 channelsof the preamble signal; generating a third sum of magnitudes of 3n+2channels of the preamble signal; determining a greatest sum of thefirst, second and third sums; and determining an integer part frequencyoffset set corresponding to the greatest sum; wherein a range of n isapproximately one-third number of channels of the preamble signal; andwherein receiving the preamble signal is receiving a preamble signaloffset by an integer part frequency offset of the integer part frequencyoffset set.
 10. The method of claim 9, wherein match filtering thepreamble signal with the plurality of pseudo noise sequences to form theplurality of filtered preamble signals comprises: match filtering thepreamble signal with each pseudo noise sequence of the plurality ofpseudo noise sequences at a plurality of delay values to formcorresponding pluralities of replicas; and combining each plurality ofreplicas to form corresponding filtered preamble signals of theplurality of filtered preamble signals.
 11. A method for generating aninteger part frequency offset set comprising: generating a first sum ofmagnitudes of sub-carriers whose index is a multiple of 3 of thepreamble signal; generating a second sum of magnitudes of sub-carrierswhose index is a multiple of 3 offset by 1 of the preamble signal;generating a third sum of magnitudes of sub-carriers whose index is amultiple of 3 offset by 2 of the preamble signal; determining a greatestsum of the first, second and third sums; and determining the integerpart frequency offset set corresponding to the greatest sum; wherein arange of n is approximately one-third number of sub-carriers of thepreamble signal.